5 GeneralizedRisk Return

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Lecture 5: Generalized Risk and Return

Readings:

Ingersoll Chapter 5 (also see chapters 8 and 9)

LeRoy & Werner Chapter 10

Merton Chapter 2 Dybvig & Ross Portfolio Efficient Sets, Econometrica, 1982

Rothschild & Stiglitz Increasing Risk I: A Definition, Journal of Economic

Theory, 1970

Rothschild & Stiglitz Increasing Risk II: Its Economic Consequence, Journal of

Economic Theory, 1971

Dybvig Distributional Analysis of Portfolio Choice, Journal of Business, 1988

Dybvig & Ingersoll Mean-Variance Theory in Complete Markets, Journal of

Business, 1982

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After having considered the special, well-known, and well (over?) used case of mean-variance

analysis, lets step back to the general problem again.

First, Risk: The CAPM makes the assumption that the variance of the return on a portfolio

measures its risk.

Consider the idea that risk is the combination of properties of a set of random outcomes that

change the evaluation of )]~([ xuE away from )(xu for concave utility u( ). Two aspects of

this definition to highlight are: (1) that what alters this evaluation (or is, on net, disliked) is, of

course, dependent upon the utility function used in the evaluation. Thought of this way, risk is

necessarily a property defined for a class of utility functions. This definition of risk also (2)conveys only the notion of dispersion of outcomes. This requires that we correct for the mean

(location) when we talk about risk. It also means that all other aspects (good and bad) of the

distribution (beyond location) are lumped into risk.

More formally

Definition: If uncertain outcomes x~

andy~

have the same location (expectation), then x~

issaid to be weakly less risky than y~

for the class of utility functions U if no individual with a

utility function in U prefers y~

to x~ . That is: )]~([)]~([ yuExuE u U. x~ isstrictly

less risky if the inequality is strict for some u U.

For some restricted classes of utility functions (some U), this ordering is complete (i.e. for allpairs of random variables x~ and y

~, with the same mean, either x~ is weakly less risky than y

~

or y~

is weakly less risky than x~ or both).

Example: Quadratic utility: as we have seen, variance is the measure of risk. Any quadratic

utility function can be written as:

u(z) = 22

zbz we restrictzto beb

z 1 so that 0)( zu

Expected utility (for any distribution for which mean and variance are defined) is written:

)(2

)]([ 22 += zbzzuE .

So, for any x~ and y~

with yx = , )]~([)]~([ yuExuE iff

22

yx .

The completeness of this ordering is unfortunately not a common property across different

(broader) classes of utility functions. Consider the example introduced by Ingersoll:

x~ = 0 with prob. = y~

= 1 with prob. = 87

4 with prob. = 9 with prob. = 81

For these simple distributions we easily calculate that: 2== yx , and2

x = 4 0, where we restrict outcomes to be bounded between zero and 2

1

)3(

c .

Expected utility is written (translating the non-central moment to the central moments):

)3()]([)]([ 2333 zzmczzEczzuE ++== ,

where 3m is the third central moment ])[(3zzE that we examined before. (Note that for

this class of utility functions variance and skewness are both disliked.)

If yx = and x~

is preferred to y~

then it must be that 0)]()(3[3322

>+ xyxy mmxc .

Thus, 323 mz + is the proper measure of risk for this class of utility functions. So, even for

classes of utility functions that imply a complete ordering of random variables, variance is not auniversal measure of risk.

For the general class of all risk averse (concave) utility functions, the ordering is obviously

incomplete. We need to find a way to reduce the scope of the problem if we are to say more, i.e.,restrictions on distributions or restrictions on utility functions (which are rarely thought to be

particularly interesting or valuable).

Mean Preserving Spreads: For illustration, lets take a look at a particular relation between x~

and y~

that allows a comparison for all u( ) in U (the class of all risk averse utility functions).

Intuitively, if we take x~ and add mean zero noise to it we should wind up with something less

attractive to risk averse agents. It turns out its not quite that simple. Why?

A random variable x~ is less risky than y~

ifg(y~

) can be obtained fromf(x~) by the

application of a series of mean preserving spreads (MPS) (wheref() andg() are density

functions).

Definition:

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elsewhere

tdxd

tdxd

tcxc

tcxc

for

for

for

for

xs

+


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Rothschild & Stiglitz Theorems on Risk:

The notion of y~

differing from x~ by the addition of a series of MPSs, implying more risk is

very intuitive, y~

has a more disperse, noisier distribution than x~ . This can be made moreprecise, more general, and less cumbersome to deal with.

Theorem 1: For the concave class of utility functions, outcome x~

is weakly less risky thanoutcome y~

iff y~

is distributed like ~~ +x where ~ is a fair game with respect to x~ .

That is: ~~~ +=xyd

and 0]~[ =xE x

Note that the law of iterated expectations tells us yx = in this case.

Proof (sufficiency): If ~~~ +=xyd

)]([ yuE )]([ += xuE equivalence of distributions]})([{ xxuEE += law of iterated expectations})]([{ xxEuE + Jensens inequality

)]([ xuE= fair game property

Given this result, we can easily see that variance is a valid measure of risk for the class of all

concave utility functions when we restrict ourselves to normal random variables.

For any b),N(~~ y and a),N(~

~ x , with b a, we can always write:

~~~ +=xyd

where a)-bN(0,~~ is independent of x~ . Since independence is

stronger than the fair game property ~ is therefore also a fair game with respect to x~ .

We can also show the related result that if y~

is riskier than x~ we know it must have a larger

variance this must be true because the quadratic utility functions are members of the concaveclass and for quadratic utility functions variance measures risk.

Alternatively: If 0]~[ =xE then, Cov 0)~,~( =x (in fact Cov 0)~),~(( =xg for all functions

g())

Therefore, ~~~ +=xyd

tells us that:

)~(yVar = )( +xVar

= ),(2)()( xCovVarxVar ++

= )()( VarxVar +

)(xVar

When location differs, we can correct by simply de-meaning and comparing xx ~ vs. yy ~

or

comparingy~

vs.).(~ xyx +

This, however, shifts around the distributions and is itselfsomewhat cumbersome and even a little odd when our goal is to describe a risk/return tradeoff.

A closely related concept to riskieris second order stochastic dominance SSD. This concept

incorporates the correction for location within the comparison.x~ displays weak 2nd order stochastic dominance over y

~if:

~~~~ ++=xy

d

where 0~

and, 0][ =+xE ,x

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Theorem 2: y~

is preferred to x~ by no strictly increasing concave utility function iff x~ SSDy~

Note: SSD incorporates the correction for location, so the class of utility functions for whichthis holds must be narrower.

Proof (sufficiency):

Theorem 1 provides )]([)]([ +++ xuExuE for all concave utility

functions.

Because is non-positive, 1st order stochastic dominance provides that:)]([)]([ + xuExuE for strictly increasing utility functions.

Thus )]([)]([ yuExuE for all increasing concave utility functions (since y~

and++x have the same distributions they have the same expected utility).

The relation between riskiness and SSD is strong but they are not identical.

If yx and xx ~ is less risky than yy ~

we know that x~ 2nd order stochastically dominates

y~ since we can write ~)(~~ ++= xyxyd

with 0)|( =xE (from riskier) so that

0))(|( =+ xyxE where xy is a degenerate non-positive random variable.

However, if x~ SSD y~

while we know that yx we cannot say that yy ~

is riskier thanxx ~ . For one thing, they may not even be ranked. Further if they are, it is possible that xx ~

is riskier than yy ~

. Let U(0,1)~~y and U(2,8)~

~x . x~ dominates y~

, and so it FSDs and

SSDs y~

, but xx ~ is riskier than yy ~

.

The difference lies in the correction for location. In SSD it is built in, we compare x~ and y~

,

while in riskier we compare xx ~ to yy ~

. An x~ with a big mean and dispersion versus ay~ with a small mean and small dispersion may be evaluated differently than xx ~ vs yy

~.

After a deviation to review the general portfolio problem and results on portfolio choice,

where this is heading is a general study of the efficient set of portfolios. SSD will play a

large role.

Table: x and y are random variables defined on [a, b] (see Ingersoll pg 123)

Concept: Utility Condition Random Variable

Condition

Distributional

Condition

Dominance )]~~([)]~~([ wyuEwxuE ++ for

any random variable w~ for

all increasing utility

~~~

+= xy

0~

Note: = notd

yx ~~

Outcome by outcome

FOSD )]~([)]~([ yuExuE for all

increasing utility functions~~~ +=xy

d

0~

F(x), G(y) are

distribution functions:

G(t) F(t) t

Prob(xt) Prob(yt)t

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SOSD )]~([)]~([ yuExuE for all

increasing concave utility

functions

~~~~ ++=xy

d

0~

0][ =+xE ,x

t

a

dssFsG )]()([

0)( t t

on average G>F

Riskier )]~([)]~([ yuExuE for all

concave utility functions ~~~

+=xy

d

0][ =xE

tt 0)(

0)( = b

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Now, Portfolio choice: To quickly review: The general portfolio problem can be cast as: an

agent chooses a portfolio to maximize his/her expected utility of returns:

)]~([ ww zuEMax ===N

i siiwzwwzz

1

~~~

s.t. 11 =w (i.e. 1= iw )

Some other restrictions we might (but wont) use include:

0w - no short sales

0~ z - limited liability for the primitive assets

1* z - non-negative wealth constraint (bankruptcy)

Form the Lagrangian:

L = )11()]~([ wzuE w +

FOCs:

=

]

~

*)

~

([ zzuEw

L

with 0

and z* = zw*

or, for each asset:

=

]~*)~([ i

i

zzuEw

Lfor all assets i

1*1 =

w

L

The concavity ofu() and linearity of the constraint implies that the FOCs are necessary andsufficient for a maximum.

The FOC must hold for the riskless asset, if it exists: = ]*)([ RzuE

So, we can subtract this from the general condition to find: 0)]~*)(([ = RzzuE i i

in the presence of a riskless asset. This will help us examine some results concerning portfolioformation in this general context.

Theorem: If a solution w* exists for a strictly concave utility function and a set of assets Zthen the probability distribution of its return is unique and if there are no redundant assets then

the portfolio w* is unique as well.

Proof: For each feasible w define new control variables s by Zw or

siiss ZwZw == Ss ,...,1= and define the function v( ), a derived utility function over

state returns:

= s siiissS ZwuuEv )()]([),...,( 1

The portfolio choice problem can then be written:

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),...( 1 SvMax s s.t. w with siiis Zw s

This is a standard convex optimization problem with an objective function that is concave in the

choice variables (note that E(u(zw)) is not concave in w) and a set of linear constraints. Thus, theoptimum (*), if it exists, is unique. The vector* describes the distribution of returns across

the states, so write:

* = Zw*

If there are no redundant assets Z has a unique left inverse ZZZL =1)( and ** Lw = is

then also unique.

We can say something specific about choices made by agents in this very general context but notmuch:

Theorem: The optimal portfolio for a strictly risk averse, non-satiated investor will be the

riskless security if and only if Rzj = j = 1, 2, , N.

Proof: When a riskless asset exists, w* satisfies the FOC: 0)]*)(([ = RzzuE j

If Rzj = j then 1* Rz = will satisfy the FOC. Since u() is strictly concave we know theprobability distribution of the return is unique. If there are no redundant assets the portfolio is

unique. (Strictly risk averse agents hold only risk free portfolios or the risk free asset. We alsosee why the strictly concave is required for uniqueness of the solution in the theorem above.)

If Rz =* we can write the FOC as 0)()( = RzERu j

0)( > Ru from strictly monotone u(), so ifz*=R is optimal, it must be that Rzj = j

This generalizes our earlier two-asset (one risky, one riskless) result.

Finally, Efficient portfolios: the efficient set of portfolios those portfolios for which there are

no other portfolios with the same or greater expected return and less risk. Alternatively, thoseportfolios which are not second order stochastically dominated.

More specifically

Definition: A portfolio w is efficient if Ew

)])}([)()(({11,

wzuEMaxwzEuUuRwEwRw

NN = =

i.e. if there exists a utility function Uu (strictly monotone, concave class), for which w solves

the investors problem:

)]([ wzuEMaxw

such that 11 =w budget constraintNRw w is a portfolio of the traded assets a restriction whose

strength is determined by the structure ofZ

We use U, the strictly monotone, concave class, because if we the use monotone, concave class

Eis trivially all NRw with 11 =w since this class allows a constant utility function, makingit uninteresting and a useless definition.

The following theorem shows that: Efficient portfolios are those for which there are no other

portfolios with the same or greater expected return and less risk (those that are not SSDed). But,

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notthe set for which there is not a less risky portfolio with the same mean return this set

conceptually includes the lower limb of the CAPM hyperbola.

Theorem: For some efficient portfolio kwith returns given byk

ez~ if

k

e

k

e zz ~ is riskier than

ww zz ~

(for any portfolio w), then wk

e zz > .

Note: It is not true in general that wk

e zz < implies that ww zz ~

is riskier thank

e

k

e zz ~ since

riskier is an incomplete ordering.

Proof: Ifk

e

k

ezz ~ is riskier than ww zz

~, then

)]~([)]~([ kewwk

e zzzuEzuE + Uu (by the definition of riskier).

If wk

e zz , then

)]~([)]~([ wk

eww zuEzzzuE + so, )]~([)]~([ w

k

e zuEzuE Uu (monotonicity).

But, this contradicts the assumption thatk

ez~ is an efficient portfolios return, so it must

be that wk

e zz > holds.

From this, we get two corollaries:

Corollary 1: Rzk

e > for all efficient portfolios since all efficient portfolios are weakly

riskier than the risk free asset.

Corollary 2: The riskier is an efficient portfolio, the higher is its expected return. In

other words, there is a risk/return tradeoff.

Now lets consider an alternative way to characterize the efficient set using a simple completemarkets example with two states (so we can draw it). Since the market is complete, we can find a

portfolio that has any distribution ofwealth across the two states that we would like. Therefore,

we can make W1 and W2 (wealth in states 1 and 2: Woz1* and Woz2*) the choice variables.

)]([)()( 2211, 21 WuEWuWuMax WW =+

subject to: p1W1 + p2W2 = Wo (budget constraint is written using the unique state prices)

L: = )()()( 22112211 WpWpWWuWu o ++

The FOCs give:

0)( 111 = pWu or,

)(11

1

Wup

=

0)( 222 = pWu or,

)( 222

Wup

=

Or,

)( s

s

ss

Wup == = state price density just as we saw before.

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Note: Under risk neutrality, state prices are proportional to the actual probabilities (or the state

price density is constant across the states). Thus, again we see that a single risk neutral agent in

the economy (who faces no restrictions on short sales or borrowing) trades to set prices in such away that is constant. Any risk averse agents in the economy (in equilibrium) will then also

trade so that )( sWu is constant across all states. Thus, their optimal choice must be to hold

riskless positions. Said another way: the risk neutral agent trades so that there is no reward forrisk bearing (all assets haveE(z)=R) so no risk averse agent bears any risk.

Examine the set of efficient (optimal) portfolios for this two state example:

W2 kWW =+ 2211

or, 12

1

2

2 Wk

W

=

slope=2

1

p

p riskless asset

oWWpWp =+ 2211

or, 12

1

2

2 Wp

p

p

WW o =

r

d Simply assume:

45 line f slope=2

1

2

1

2

1

p

p>

W1

Line with slope =2

1

p

p is the budget constraint

Line with slope =2

1

is a line of constant expected wealth

Draw an indifference curve for an arbitrary risk averse utility function. How do we know that

the constant expected wealth line is tangent to the indifference curve at the 45 line?

It must be from routward on the line, you stay at the same expected wealth but add risk if you

move in either direction. Thus, the constant expected wealth line through rmust be tangent as itmust be below the indifference curve away from ron either side.

Which of the possible consumption bundles will be chosen by a rational agent?

Consider point din the picture. By concavity of the utility function, dis no better than r

(same expected wealth but more risk) and by strict monotonicityfis dominated by dsince dhas largerW1 and W2. So, by analogy all points on the budget line below rare dominated by

r, the riskless asset (they all give less expected wealth and more risk).

Rational choices are above the point ron the budget line in this picture. Thus, only choices

with W2 W1 are optimal. Moving in this direction gets more risk and higher (not lower)

expected wealth. Where will the indifference curve be tangent to the budget constraint?

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Depends on u but we know it will be above point rbecause we assumed the budget line had a

steeper slope than the constant expected value line (that is,2

1

2

1

2

1

2

1

>>

p

p

p

por,

2

2

1

1

pp> or, 21 > ).

Thus, the state price per unity probability of wealth in state one is greater than in state 2.

Choosing W2 W1 is, in this sense, a cost minimizing choice (for a given distribution of

wealth, choosing W2 W1 has lower cost than the reverse). Given state independent vN-Mutility we will show later that cost minimization in this way is really the only restriction

imposed by maximizing behavior.

We can also find the result: if1 > 2 then W2 W1 from the FOC of the investors decisionproblem in this example:

ssWu = )(

So, if1 > 2 we require )()( 21 WuWu > , or, since u is weakly decreasing, 21 WW .

We can generalize this two state example:

Theorem (Dybvig J of B 88): Assume a complete market and equally probable states. For the

strictly monotone, concave class of utility functions U, w is an efficient portfolio if and only if,

for all states r, s:

rssr ZwZw >

Proof (sketch): Suppose w is an efficient portfolio, then it follows from the FOC (

))( ssZwu = and the concavity ofu() that rssr ZwZw > (the 2nd inequality is weak

since we dont require strict concavity). The contra-positive of this is: srsr ZwZw>

.Now suppose that srsr ZwZw > or equivalently rssr ZwZw > . Graphically this

says:

ZwChoose any function g() which hasg(Zws)=s fors > 0. We know that this function is

positive and weakly decreasing so re-label it .)( ssZwu =

Integrate this function and you get a strictly monotone concave function u() such that it satisfies

the FOC at the candidate w. Note that we have found a u() scaled so that the Lagrangemultiplier equals 1. But, we can always do this since u() is unique only up to a positive

affine transformation. The constant of integration from the step uu is the rest of this

transformation. This is the equivalence between maximizing behavior and cost minimization.

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Problem: Look back to the two state complete markets example. Prove that if state prices

are proportional to the actual probabilities then any choice not on the 45 line is riskier than abundle on the 45 line.

Instead of the formal proof of sufficiency for the last theorem, to provide a little intuition, look atthe situation of a complete market where Ss

1= s=1,,S (an unnecessary simplification).

Now, since there are S states, there are S! ways to order or assign the lottery outcomes to states

(which state generates greater wealth than which). This means that there is some cheapest way

to order the lottery. Suppose that one of the cheapest ways does not assign outcomes in reverse

order to the state price density. Then states r, s such that sr > but sr ww > . Now, switchthe outcomes for states rands. The change in cost is:

2))(())(()()( Swwppwwwpwpwpwp srrssrrsssrrrssr ==++

which is a negative number implying a cost decrease with no change in expected utility for

state independent utility of wealth and so a contradiction.

Example Verifying the efficiency of a given portfolio:

The last theorem can be generalized and Ingersoll uses the general result to build a numerical

example illustrating the idea that verifying the efficiency of a given portfolio requires only thatwe check the ordering of returns it assigns across states. It is, therefore, also true that all

portfolios which impose the same orderings on their returns as that of an efficient portfolio are

efficient, i.e., some strictly increasing concave utility function sees that portfolio as optimal.

If the matrix Z has no redundant assets then the rank ofZ is NS.When N = S the market is complete and thus there is a unique vector that supports Z. Thusthere is only one ordering of the state contingent payoffs that can represent an efficient portfolio.

If N < S, the market is incomplete, is not unique (there are N equations in S unknowns in thesupporting equation) so not all efficient portfolios need have the same orderings of returns. Anexample from the text will help illustrate this here it has been changed by putting things in

terms of thes where the book uses marginal utilities see the FOC for the translation:

Consider the market characterized by:

=

6.00.3

5.12.1

4.26.0

Z where3

1321===

S=3 3! or 6 potential orderings of returns. With 2 assets, only 4 orderings are feasible.

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Thefeasible set is:

w . So, any such portfolio is optimal forsome agent. Of the feasibleorderings, only (3, 2, 1) ( 3

11 and the set with 32 zz if 23 > etc. All these intersected with the

set of marketed assets. We then have the following:

Theorem: IfMis a complete market thenEis a convex set.

Proof: Eis the intersection of convex sets and so is convex.

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Theorem: In a risk neutral marketE=MsoEis necessarily convex.

Proof: In this case s is constant across states. Thus, all orderings of returns across states are

efficient. From the FOC we know srsr zz < , but also srrs zz < .

Thus, both orderings are efficient if sr = .Alternatively: All assets have the same expected return and so all possible portfolios may be held

by maximizing risk neutral agents.

In general, ).( EME = This second set is the union over all valid s of E .

That is, }{ll srl

zzzE =

where (r, s) = (rl, sl) for some l ll sr


17/20

Let 41

4321==== Let Z =

485050

483852

485244

728266

The valid set of price vectors is: )}39,40,28,21(),59,40,68,1{(/ }0{4 spanR +

We can divide by 8,088 or 6,648 respectively to get 1=pZ .

Thus, we have two orderings onp and, since ss = 41 , on.

Look at 11 Zwz = 22 Zwz = with )0,0,1(1 =w and )0,1,0(2 =w

Since )50,52,44,66(1 =z has an opposite ordering to (1, 68, 40, 59) and

)50,38,52,82(2=z has an opposite ordering to (21, 28, 40, 39) both asset 1 and asset 2 are

efficient portfolios.

However, )50,45,48,74(221

121

+ ww is not efficient since it isnt in the opposite order toeither valid p ().

Alternatively, consider )4.2,4.,1(* =w . Because vN-M agents view equally probable statessymmetrically, we know that for every strictly monotone vN-M (state independent) utility

function that:*)]([ zwuE )]2.45,48,4.50,74([uE=

)]4.50,2.45,48,74([uE= (state independent utility))]50,45,48,74([uE> (strictly monotone utility)

)]([ 221

121 wwuE +=

Thus, a convex combination of two efficient portfolios w1 and w2 is not an efficient portfolio

since it is dominated by another portfolio for every monotone state independent utility agent.

Thus,Eis not a convex set.

2w

Picture: E is the shaded area

221

121 ww +

1w

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Systematic & Non-Systematic Risk

How do we understand these concepts using the Rothschild-Stiglitz notion of riskiness and

our definition of efficiency?

Analogous to the CAPM and Beta, we have seen that the way in which the risk or variabilityof an assets returns affects the risk/expected return of an individuals optimal portfolio isthrough its correlation with the state price density (rather than its correlation with Zm.)

o Equivalently, the correlation between an investors marginal utility of the return

on his/her optimal portfolio and any assets return.

That is, if an assets returns have an inverse ordering across the states of nature as does the

marginal utility of z* () then it has a similar correlation with the marginal utility of z* as

does z* itself and much of that assets variability contributes to the value of the portfolio

think of CAPM and correlation with the market return.

Systematic risk is the notion of an individual assets contribution to the risk of an efficientportfolio (i.e. what portion of an assets risk is priced).

Consider marginal utility of returns as a random variable: )~(~k

ek zuu =

Now, consider a conceptual regression: ikkikiki uz ~~~ ++=

which defines the systematic risk and non-systematic risk of asset i with respect to efficient

portfolio kand utility function u( ).

Consider)(

)~,(

k

ikik

uVar

zuCov

=

Normalize ik by)(

)~,(

k

k

ekkk

uVar

zuCov

= to remove any influence of the scale of the utility function

we are using: so define)~,(

)~,(k

ek

ik

kk

ik

ikzuCov

zuCovb

==

.

ikb provides us with a measure of the systematic risk of asset i with respect to efficient portfolio

kthat is independent of the scale of the utility function, u().

This measure possesses a portfolio property that the b of a portfolio is the weighted average

of the bs of the assets in the portfolio (using the portfolio weights)

The ordering asset i is riskier than assetj by this measure is a complete ordering and the

ordering is independent of the efficient portfolio chosen. Therefore, if we can identify an

efficient portfolio and measure marginal utility we could correct the problem we had beforeof an incomplete ordering on riskiness.

If there is a riskless asset, we can write excess expected returns as being proportional to b.

Rearrange the FOC to write:

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][][ kik uREzuE = now rewrite the left hand side and rearrange the equation

)]([][][],[ RzuEzuEuREzuCov ikikkik ==

This holds for all assets, includingk

ez , so:

)(

)(

),(

),(

Rz

Rz

zuCov

zuCovb

k

e

i

k

ek

ik

ik

=

= or, )( RzbRz

k

eiki +=

And, since we know Rzk

e > , Rzi is positively proportional to ikb .

Suppose there is no riskless asset What is the equivalent of a zero-beta asset here?

The relation can come more quickly from the standard E[ zi] = 1 for all assets i.

Our problem is b is in terms of ku , which is not something we can easily measure. In order to

see the generality of this result is lets examine some special (familiar) cases.

If utility is quadratic, u is linear ink

ez~ .

Then, ikb becomes ikke

i

k

eik

zVar

zzCovb ==

)~(

)~,~(. Then, all we need is the efficiency of the

market portfolio. We get that from the assumed quadratic utility since, as we have seen and

will see again, it generates 2fs which impliesEis convex. So, the market portfolio is inE.

If asset returns are multivariate normal

Steins lemma says )~,~cov())~(()~),~(( ik

e

k

ei

k

e zzzuEzzuCov =

So, ikb again becomes)~(

)~,~(k

e

ike

ikzVar

zzCovb = . See comment above. Normality also implies

2fs.

The consumption CAPM (Breeden) spills out of this, as well. What we are seeing is that it

all depends on what is a sufficient statistic for marginal utility or . With quadratic utility or

multivariate normal returns, the return on an efficient portfolio is sufficient for )(k

ezu . The

assumptions of the CCAPM imply aggregate consumption is a sufficient statistic for

marginal utility.

Nonsystematic Risk:Consider the ik from our conceptual regression. ik depends upon both the benchmark

portfolio and the utility function chosen. Thus, the ik we identified is not an unequivocal

measure of nonsystematic risk that will be agreed upon by all investors.

The one exception is a complete market or an effectively complete market. There we know thatthe marginal utilities of all investors are exactly proportional ( is unique), thus for all efficient

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portfolios and all utility functions the ku s are perfectly correlated and ijik = for all investors.

(In a pareto efficient market all investors see the same state prices, so no valuable trades can becreated. Thus, the marginal utilities must all be proportional.)

Sufficient conditions for -risk to be nonsystematic if it is uncorrelated with the market return

(not true generally) are a pareto efficient market and that is a fair game with respect to mz . That is, if we write iz

~as iii xz

~~~ += where 0]~[ =mi zE and that the market is effectively

complete (which implies thatEis convex), then i is unequivocally nonsystematic risk.

ix~

may be systematic, nonsystematic, or a combination of both types of risk. The market is

(effectively) complete soEis convex and the market portfolio is efficient. There exists,

therefore, a mu for which mz is the efficient portfolio and from the pareto efficiency of the

market we know )( mmk zuau = for all investors k. Since is a fair game with respect to mz

it is also uncorrelated with )( mm zu and so uncorrelated with any ku . is thereforerecognized as nonsystematic risk by all investors and will have no price.

In particular models the notion of nonsystematic risk being risk that is uncorrelated with the stateprice density can be a handy representation.

5 GeneralizedRisk Return

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